Читать книгу Quantum Mechanics, Volume 3 - Claude Cohen-Tannoudji - Страница 46

The creation and annihilation operators introduced in this chapter lead to compact and general expressions for operators acting on any particle number N. These expressions involve the occupation numbers of the individual states but the particles are no longer numbered. This considerably simplifies the computations performed on “N-body systems”, like N interacting bosons or fermions. The introduction of approximations such as the mean field approximation used in the Hartree-Fock method (Complement D_XV) will also be facilitated.

We have shown the complete equivalence between this approach and the one where we explicitly take into account the effect of permutations between numbered particles. It is important to establish this link for the study of certain physical problems. In spite of the overwhelming efficiency of the creation and annihilation operator formalism, the labeling of particles is sometimes useful or cannot be avoided. This is often the case for numerical computations, dealing with numbers or simple functions that require numbered particles and which, if needed, will be symmetrized (or antisymmetrized) afterwards.

In this chapter, we have only considered creation and annihilation operators with discrete subscripts. This comes from the fact that we have only used discrete bases or {|ui〉} or {|vj〉} for the individual states. Other bases could be used, such as the position eigenstates {|r〉} of a spinless particle. The creation and annihilation operators will then be labeled by a continuous subscript r. Fields of operators are thus introduced at each space point: they are called “field operators” and will be studied in the next chapter.